Classification of nonorientable regular embeddings of Hamming graphs

TL;DR

This paper classifies nonorientable regular embeddings of Hamming graphs, identifying specific parameter conditions for their existence and providing constructions and descriptions of these embeddings.

Contribution

It provides a complete classification of nonorientable regular embeddings of Hamming graphs, including existence conditions and explicit constructions.

Findings

Existence of nonorientable regular embeddings only for specific parameters

Explicit constructions of these embeddings are provided

The classification covers all cases for Hamming graphs with given parameters

Abstract

By a regular embedding of a graph K in a surface we mean a 2-cell embedding of K in a compact connected surface such that the automorphism group acts regularly on flags. In this paper, we classify the nonorientable regular embeddings of the Hamming graph H(d,n). We show that there exists such an embedding if and only if n=2 and d=2, or n=3 or 4 and d>0, or n=6 and d=1 or 2. We also give constructions and descriptions of these embeddings.